Although most examples of singular hyperbolic attractors in the literature satisfy conditions (SH3) - (SH7) (see the examples in [1]}, [2]}, [3]}, [4]}, [5]}), there exist singular hyperbolic attractors that do not satisfy these conditions and admit infinitely many ergodic components. For this reason, these conditions are necessary for our statement to be true in full generality. For example, one can construct a family of Lorenz-type attractors whose singular sets have infinitely many components, and these examples admit countably many SRB measures, but these maps are not topologically transitive (see Section REF ).
